Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 3.3.6.10. Let $X$ be a Kan complex. Then the comparison map $\rho _{X}^{\infty }: X \rightarrow \operatorname{Ex}^{\infty }(X)$ is a homotopy equivalence.

Proof. Since $\operatorname{Ex}^{\infty }(X)$ is also a Kan complex (Proposition 3.3.6.9), it will suffice to show that $\rho _{X}^{\infty }$ is a weak homotopy equivalence (Proposition 3.1.5.12), which follows from Proposition 3.3.6.7. $\square$